A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is?

Options

A. 0.1
B. 0.15
C. 0.2
D. 0.25

Answer

To find the percentage of water in the mixture that a dishonest milkman uses to gain a profit of 25%, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Profit Percentage**:
The profit percentage is given by the formula:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 \]
Here, the milkman gains a profit of 25%.

2. **Setting Up the Equation**:
Let the cost price of the milk be \( CP \). If the milkman sells the milk at cost price but mixes it with water, we can express the profit as:
\[ \frac{\text{Profit}}{CP} \times 100 = 25 \]
This implies:
\[ \frac{\text{Profit}}{CP} = \frac{25}{100} = \frac{1}{4} \]

3. **Calculating the Effective Selling Price**:
If the profit is \( \frac{1}{4} \) of the cost price, then:
\[ \text{Selling Price (SP)} = CP + \text{Profit} = CP + \frac{1}{4}CP = \frac{5}{4}CP \]

4. **Understanding the Mixture**:
The milkman is mixing milk with water. Let’s assume he sells 1 liter of the mixture. If he gains a profit of 25%, it means he is selling 1 liter of mixture for \( \frac{5}{4}CP \).

5. **Finding the Ratio of Milk to Water**:
If we denote the amount of milk in the mixture as \( x \) liters and the amount of water as \( y \) liters, we have:
\[ x + y = 1 \quad \text{(total mixture)} \]
The cost price of the milk is \( CP \) per liter, and since he is selling at cost price, the effective cost of the mixture is:
\[ \text{Cost of mixture} = x \cdot CP + y \cdot 0 = x \cdot CP \]
The selling price of the mixture is \( \frac{5}{4}CP \).

6. **Setting Up the Equation for Cost**:
Since he sells the mixture at \( \frac{5}{4}CP \), we equate:
\[ \frac{5}{4}CP = x \cdot CP \]
Dividing both sides by \( CP \) (assuming \( CP \neq 0 \)):
\[ \frac{5}{4} = x \]

7. **Finding the Amount of Water**:
Since \( x + y = 1 \):
\[ y = 1 - x = 1 - \frac{5}{4} = -\frac{1}{4} \]
This indicates that the milkman has added water to increase the volume.

8. **Calculating the Percentage of Water**:
The percentage of water in the mixture is given by:
\[ \text{Percentage of Water} = \frac{y}{x + y} \times 100 \]
Since \( x = \frac{4}{5} \) and \( y = \frac{1}{5} \):
\[ \text{Percentage of Water} = \frac{y}{1} \times 100 = \frac{1}{5} \times 100 = 20\% \]

9. **Final Answer**:
Therefore, the percentage of water in the mixture is:
\[ \text{Percentage of Water} = 20\% \text{ or } 0.2 \]